Fill a large pot with water, and add beets. Bring to a boil over medium-high heat, and cook for about 30 minutes. Add potatoes to the water, and continue to cook for another 30 minutes. Add carrots, and continue boiling until everything is soft, about 15 minutes. Drain, and let the vegetables cool until cool enough to handle.
Cut vegetables into cubes, and place them in a serving bowl. Stir in mayonnaise and mustard until everything is well coated. Serve warm.